Assume we have this strictly convex quadratic programming:
$$f(x) = x^\top A x + b^\top x,$$ $$Ax \leq b$$ $$ 0 \leq x \leq 1$$
Where $A$ is symmetric and positive definite, and the feasible set is nonempty. Does strong duality and Slater's condition holds in this case.
The feasible set is nonempty and compact. The objective is continuous. So by Weirstrass we have a finite optimal value. The constraints and the objective are all convex functions. Finally, since the feasible set is nonempty, we have a vector which satisfies the linear inequalities. Thus we have Slater's condition (linear inequalities do not require strict feasibility). So we also have strong duality.